All of the 5th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.00$ each for students, and the group paid $$50.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$11.00$ each for teachers and $$12.50$ each for students, and the group paid $$131.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4y = 50}$ ${11x+12.5y = 131.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-11x-8y = -100}$ ${11x+12.5y = 131.5}$ Add the top and bottom equations together. $ 4.5y = 31.5 $ $ y = \dfrac{31.5}{4.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {5.5x+4y = 50}$ to find $x$ ${5.5x + 4}{(7)}{= 50}$ $5.5x+28 = 50$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {11x+12.5y = 131.5}$ and get the same answer for $x$ ${11x + 12.5}{(7)}{= 131.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.